Divisors of expected Jacobian type

Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.